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Physics
Special and General Relativity
Understand Twin Paradox, Leading Clocks Lag & Doppler Effect
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[QUOTE="PeroK, post: 6071579, member: 493650"] Mathematics can help here. Imagine two clocks a distance ##L## apart in their rest frame. A light source half-way between the clocks sends out a pulse in both directions and illuminates the clocks simultaneously. If the clocks are synchronised in their rest frame, then the clocks read the same time when they are briefly illuminated. Let's call this reading ##T##. Now, if this setup is moving with speed ##v## to the right in a second reference frame, then the clocks will not be illuminated at the same time. The rear clock will be illuminated first (and show time ##T##), then the leading clock will be illuminated some time later (and show time ##T##). Hence the leading clock lags in this frame. How much is this lag? In the second frame, from length contraction, the light source is ##\frac{L}{2\gamma}## from both clocks. The pulse reaches the rear clock in a time ##t_1 = \frac{L}{2\gamma (c+v)}##, as (in this frame) the light and the rear clock are moving towards each other. The pulse reaches the front clock in a time ##t_2 = \frac{L}{2\gamma (c-v)}##. The difference is: ##\Delta t = t_2 - t_1 = \frac{L}{2\gamma}(\frac{1}{c-v} - \frac{1}{c+v}) = \frac{L}{2\gamma}(\frac{2v}{c^2-v^2}) = \frac{L}{\gamma}(\frac{v}{c^2})\gamma^2 = \frac{\gamma Lv}{c^2}## Hence, in the second frame, the leading clock is illuminated a time ##\frac{\gamma Lv}{c^2}## after the rear clock. Finally, both clocks are time dilated, so when the leading clock is illuminated, the rear clock has advanced from ##T## to ##T + \frac{\Delta t}{\gamma} = T + \frac{Lv}{c^2}##. That, then, gives us simultaneous readings for the two clocks in the second frame: The rear clock reads ##T + \frac{Lv}{c^2}## when the leading clock reads ##T##. [/QUOTE]
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Physics
Special and General Relativity
Understand Twin Paradox, Leading Clocks Lag & Doppler Effect
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